Question

Suppose that the sound level of a conversation is initially at an angry 71 dB and then drops to a soothing 53 dB. Assuming that the frequency of the sound is 506 Hz, determine the (a) initial and (b) final sound intensities and the (c) initial and (d) final sound wave amplitudes. Assume the speed of sound is 347 m/s and the air density is 1.21 kg/m³.

Answer 1

The intensity level is given by

β = 10*log(I/I0)

I is the intensity of the sound produced

I₀ is the threshold of hearing = 10^-6 μW/m²

Given initial level of sound β_i = 71 dB

final sound level β_f = 53 dB

71 = 10*log(I_i/10^-6)

initial intensity of the sound I_i = 12.589 μW/m²

53 = 10*log(I_f/10^-6)

I_f =0.19953  μW/m²

The intensity of the sound is related to the amplitude by the equation

I = 0.5*ρvω²S²

For the initial sound level

12.589*10^-6 = 0.5*1.21*347*(2*pi*506)^2*S_i²

S_i = 77.023 nm   

For the final sound level

0.19953*10^-6 = 0.5*1.21*347*(2*pi*506)^2*S_f²

s_f = 9.69686 nm

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